I am trying to prove the following statement for a diffusion $S_t$ with $S_0=0$ and a real function $f$ that is continuous at $0$:
$$\lim_{t\rightarrow 0} \mathbb E f(S_t) = f(0), \text{ if } \mathbb ...

Kolmogorov's extension theorem allows for the construction of a variety of measures on infinite-dimensional spaces, and its conditions are supposedly "trivially satisfied by any stochastic process". ...

The Ito's formula stated in most books in stochastic calculus is in the form $F(t,X_t)$, where $F: \mathbb{R}^{d+1} \rightarrow \mathbb{R}$ is a $d+1-$dimensional deterministic $C^{1,2}$ function and ...

Consider two continous-time stochastic processes $\{A(t)\}_{t \ge 0}$ and $\{B(t)\}_{t \ge 0}$ with $A(t)=t$ and $B(t)=t$. Each process starts at $t=0$ and emits "ticks" at increasing time slots. For ...

The following question arose while I was trying to explore possible further extensions of a CLT by Liverani which I mentioned here already (see this link, I can tell you more details upon request). It ...

Could you point out a comprehensive reference book (or more than one, if it is the case) on Quantum Probability that introduces the subject and then gradually builds up to the edges of contemporary ...

in the definition of the weak GARCH processes they use the terminology of the 4th-order stationarity of the process $(X_t)$. I know the definition of 2n-order stationarity, but I'm not exactly sure, ...

Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F:
$$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$
In this case, $F^{-1}$ is the (generalized) inverse of ...

For $X$ a measurable space and $P,Q$ two probability measures on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as
$$D(Q\|P)=\int_X \log(\frac{dQ}{dP})dQ,$$
...

Let $C$ be a Jordan curve in $\mathbb{R}^2$. By the Jordan curve theorem, $\mathbb{R}^2 \smallsetminus C$ is uniquely partitioned into two connected regions $A$ and $B$ (the interior and exterior).
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